10-2 Regression 535 Let’s consider the nine pairs of jackpot>ticket data included in Table 10-1 from the Chapter Problem. Those nine pairs of data result in this regression equation: yn = -10.9 + 0.174x (as shown in Examples 1 and 2). The slope of 0.174 tells us that if we increase the jackpot x by 1 (million dollars), the predicted number of tickets sold will increase by 0.174 million (or 174,000 tickets). That is, for every additional 1 million dollars added to the jackpot amount, we expect the ticket sales to increase by 174,000 tickets. This realization has led lottery officials to adjust their rules to make winning more difficult so that jackpots will grow considerably larger and drive greater lottery ticket sales. Outliers and Influential Points A correlation>regression analysis of bivariate (paired) data should include an investigation of outliers and influential points, defined as follows. DEFINITIONS In a scatterplot, an outlier is a point lying far away from the other data points. Paired sample data may include one or more influential points, which are points that strongly affect the graph of the regression line. To determine whether a point is an outlier, examine the scatterplot to see if the point is far away from the others. Here’s how to determine whether a point is an influential point: First graph the regression line resulting from the data with the point included, then graph the regression line resulting from the data with the point excluded. If the regression line changes by a considerable amount, the point is influential. Consider the nine pairs of jackpot>ticket data from Table 10-1 in the Chapter Problem. The scatterplot located to the left below shows the regression line. If we include the additional pair of x = 980 and y = 12, we get the regression line shown to the right below. The additional point 1980, 122 is an influential point because the graph of the regression line did change considerably in the right graph. Compare the two graphs to see clearly that the addition of this one pair of values has a very dramatic effect on the regression line, so that additional point is an influential point. The additional point is also an outlier because it is far from the other points. CP EXAMPLE 5 Influential Point Original Jackpot , Ticket Data from Table 10-1 Jackpot , Ticket Data with Additional Point: (980, 12)
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